Properties

Label 2-370-185.104-c1-0-11
Degree $2$
Conductor $370$
Sign $0.959 + 0.282i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.434 + 1.19i)3-s + (0.766 − 0.642i)4-s + (−1.33 − 1.79i)5-s + 1.27i·6-s + (2.39 − 0.422i)7-s + (0.500 − 0.866i)8-s + (1.06 + 0.890i)9-s + (−1.86 − 1.22i)10-s + (−0.539 + 0.934i)11-s + (0.434 + 1.19i)12-s + (4.50 − 3.77i)13-s + (2.10 − 1.21i)14-s + (2.72 − 0.813i)15-s + (0.173 − 0.984i)16-s + (3.03 + 2.54i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.250 + 0.689i)3-s + (0.383 − 0.321i)4-s + (−0.596 − 0.802i)5-s + 0.518i·6-s + (0.905 − 0.159i)7-s + (0.176 − 0.306i)8-s + (0.353 + 0.296i)9-s + (−0.590 − 0.388i)10-s + (−0.162 + 0.281i)11-s + (0.125 + 0.344i)12-s + (1.24 − 1.04i)13-s + (0.563 − 0.325i)14-s + (0.703 − 0.210i)15-s + (0.0434 − 0.246i)16-s + (0.734 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88663 - 0.272022i\)
\(L(\frac12)\) \(\approx\) \(1.88663 - 0.272022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (1.33 + 1.79i)T \)
37 \( 1 + (-5.80 + 1.82i)T \)
good3 \( 1 + (0.434 - 1.19i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.39 + 0.422i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.539 - 0.934i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.50 + 3.77i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.03 - 2.54i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.238 - 0.656i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (4.52 + 7.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.50 - 2.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.83iT - 31T^{2} \)
41 \( 1 + (4.58 - 3.85i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 + (6.62 - 3.82i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.71 + 1.36i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (8.83 + 1.55i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.74 + 9.22i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-4.03 + 0.710i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-10.0 - 3.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 0.0962iT - 73T^{2} \)
79 \( 1 + (0.293 - 0.0516i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (9.17 - 10.9i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-12.0 - 2.12i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.60 + 4.50i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.20417048383027028329675896336, −10.70172098524521016838422652791, −9.833282167558984500710670658750, −8.294655479417074569735112044588, −7.925714575771802517017110630973, −6.23458798602523536401362003187, −5.07255580839138627104553710577, −4.50411110616624736374182894716, −3.48746769366147292267534323553, −1.43507473271096749585971664760, 1.68876000513441133483854427764, 3.38293790656414954238677931827, 4.39409280606608460230005167476, 5.83228429966135915861570031440, 6.61029498332838093398353452810, 7.56563507962891955512415500998, 8.191404202156868607506469562295, 9.657228261578675675243627821869, 11.01239769876535591694299512348, 11.68082011933766602411593960670

Graph of the $Z$-function along the critical line